Let us survey the related mathematical results.
Time periodic solutions are widely studied for other differential equations. For example, Matsumura and Nishida [1]} investigates those of the compressible Navier Stokes equation.
On the other hand, as far as conservation laws concerned, it has not been received much attention until now. Takeno [2]} studies a single conservation law and proved the existence of a time periodic solution for the space periodic boundary condition.
The key tool is the decay estimate in Tadmor [3]}.
It should be noted that we cannot apply the method of [3]} to systems. Greenberg and Rascle [5]} treat with an artificial system of conservation laws by
the Glimm scheme. Although the existence theorem for isentropic gas dynamics is recently obtained in Tsuge [6]}, the initial and boundary conditions are restrictive.
